Optimal. Leaf size=22 \[ \frac{2 \tan ^{-1}\left (\frac{2 \log (7 x)+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0180555, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{2 \log (7 x)+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\log (7 x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \log (7 x)\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{1+2 \log (7 x)}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0285106, size = 22, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 \log (7 x)+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 20, normalized size = 0.9 \begin{align*}{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,\ln \left ( 7\,x \right ) \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.72742, size = 76, normalized size = 3.45 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \log \left (7 \, x\right ) + \frac{1}{3} \, \sqrt{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.164534, size = 22, normalized size = 1. \begin{align*} \operatorname{RootSum}{\left (3 z^{2} + 1, \left ( i \mapsto i \log{\left (\frac{3 i}{2} + \log{\left (7 x \right )} + \frac{1}{2} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.79949, size = 26, normalized size = 1.18 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \log \left (7 \, x\right ) + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]